Chapter 6 Free quantum electromagnetic field

This chapter studies the canonical quantization of the electromagnetic field and explains how to deal with the gauge symmetry of this system.

6.1 Classical electromagnetism

Let us start by recalling some aspects of classical electromagnetism in \(d=4\) spacetime dimensions before moving to the general case of \(d\) dimensions. Maxwell’s theory of electromagnetism (in \(d=4\)) is traditionally formulated in terms of the electric field \(\mathbf {E}=(E_1,E_2,E_3)\) and the magnetic field \(\mathbf {B}=(B_1,B_2,B_3)\), whose dynamics is governed by Maxwell’s equations

\begin{flalign} \nabla \cdot \mathbf {E} &= \rho \quad ,\qquad \nabla \times \mathbf {E} = -\frac {\partial \mathbf {B}}{\partial t}\quad ,\\[5pt] \nabla \cdot \mathbf {B} &= 0\quad ,\qquad \nabla \times \mathbf {B} = \mathbf {j} + \frac {\partial \mathbf {E}}{\partial t} \quad . \end{flalign} The charge density \(\rho \) and the current density \(\mathbf {j}=(j^1,j^2,j^3)\) are required to satisfy the charge conservation law

\begin{flalign} \label {eqn:chargeconservationMax} \frac {\partial \rho }{\partial t} + \nabla \cdot \mathbf {j} =0\quad . \end{flalign} Recalling that every divergence-free vector field is the curl of another vector field, we can solve the homogeneous Maxwell equation \(\nabla \cdot \mathbf {B} =0\) and write without loss of generality

\begin{flalign} \mathbf {B} = \nabla \times \mathbf {A}\quad . \end{flalign} Inserting this into the other homogeneous Maxwell equation and recalling that every curl-free vector field is the gradient of a function, we can write

\begin{flalign} \mathbf {E} = \nabla \phi -\frac {\partial \mathbf {A}}{\partial t}\quad . \end{flalign} The tuple \((\phi ,\mathbf {A}) = (\phi ,A_1,A_2,A_3)\) is called the electromagnetic potential and it can be used as an alternative set of variables, in contrast to the traditional variables \(\mathbf {E}\) and \(\mathbf {B}\), to formulate electromagnetism. Since in these variables the homogeneous Maxwell equations are automatically solved (by construction), we are left with the two inhomogeneous equations

\begin{flalign} \label {eqn:Maxwellpotentiald=4} \nabla ^2\phi -\frac {\partial }{\partial t} \big (\nabla \cdot \mathbf {A}\big ) \,=\, \rho \quad ,\qquad \frac {\partial ^2 \mathbf {A}}{\partial t^2} - \nabla ^2\mathbf {A} + \nabla \big (\nabla \cdot \mathbf {A}\big ) - \frac {\partial }{\partial t}\nabla \phi \,=\, \mathbf {j}\quad . \end{flalign} The variables \((\phi ,\mathbf {A})\) turn out to be more fundamental than the \((\mathbf {E},\mathbf {B})\) variables: As we will see later, it is the electromagnetic potential \((\phi ,\mathbf {A})\) that couples to a charged Dirac field and not the \((\mathbf {E},\mathbf {B})\) fields. There are further arguments why the electromagnetic potential is more fundamental and I recommend you to google e.g. the “Aharonov-Bohm effect”. However, there is a price to pay: The defining equations (6.3) do not fix \((\phi ,\mathbf {A})\) uniquely for a given \((\mathbf {E},\mathbf {B})\), so that one has to deal with redundancies in the description. Indeed, you can easily check that the potentials

\begin{flalign} \label {eqn:Abeliangaugecomponents} \big (\phi ,\mathbf {A}\big )~\sim ~\Big (\phi + \frac {\partial \alpha }{\partial t}, \mathbf {A} + \nabla \alpha \Big ) \end{flalign} define the same \(\mathbf {E}\) and \(\mathbf {B}\) fields, for any choice of function \(\alpha (x)\) on the Minkowski spacetime. Hence, such electromagnetic potentials should be regarded as being physically equivalent, which can be formalized mathematically by working with an equivalence relation on the set of potentials. We will come back to this issue later in this section.

The equations (6.4) do not look very pleasant and it is also not obvious if they are covariant under Poincaré transformations. These two issues can be solved by introducing the relativistic electromagnetic potential \(A_\mu \) in terms of the covector field

\begin{flalign} A := \begin{pmatrix} A_0\\A_1\\A_2\\A_3 \end {pmatrix} :=\begin{pmatrix} \phi \\A_1\\A_2\\A_3 \end {pmatrix} \end{flalign} and the relativistic current \(j^\mu \) in terms of the vector field

\begin{flalign} j := \begin{pmatrix} j^0\\j^1\\j^2\\j^3 \end {pmatrix} :=\begin{pmatrix} \rho \\j^1\\j^2\\j^3 \end {pmatrix} \end{flalign} on the Minkowski spacetime \(\bbR ^4\). Let us further define the field strength tensor by antisymmetrized partial differentiation

\begin{flalign} F_{\mu \nu }:=\partial _\mu A_\nu - \partial _\nu A_\mu \quad . \end{flalign} A quick check shows that the matrix represented by \(F_{\mu \nu }\) reads as

\begin{flalign} F = \begin{pmatrix} 0 &-E_1 & -E_2&-E_3 \\ E_1&0&B_3&-B_2\\ E_2&-B_3&0&B_1\\ E_3&B_2&-B_1&0 \end {pmatrix}\quad , \end{flalign} i.e. it packages the \(\mathbf {E}\) and \(\mathbf {B}\) fields into a single \((0,2)\)-tensor field, and that the equation

\begin{flalign} \partial _\mu F^{\mu \nu } = -j^\nu \end{flalign} captures the inhomogeneous Maxwell equations. When written in terms of \(A_\mu \), the latter equation reads as

\begin{flalign} -\partial ^2 A_\mu + \partial _\mu \big (\partial _\nu A^\nu \big ) = j_\mu \quad , \end{flalign} where we recall that \(\partial ^2 = \eta ^{\mu \nu }\partial _\mu \partial _\nu \). The components of this equation are precisely (6.4), hence we found a neat way to write everything in a manifestly Poincaré covariant form.

It is now obvious how to generalize electromagnetism to the \(d\)-dimensional Minkowski spacetime \((\bbR ^d,\eta )\): We simply introduce a \(d\)-dimensional covector field \(A_\mu \) for the electromagnetic potential, a \(d\)-dimensional vector field \(j^\mu \) satisfying \(\partial _\mu j^\mu =0\) for the current and demand the equation of motion (6.10) to hold on \(\bbR ^d\). A nice feature of this equation of motion is that it is the Euler-Lagrange equation of the following action functional

as one can easily check. For a trivial current \(j^\mu =0\), i.e. for electromagnetism in vacuum, this action is clearly invariant under (active) Poincaré transformations

\begin{flalign} T\,:\, A_{\mu }(x) \,&\longmapsto \, (TA)_\mu (x) =\Lambda _{\mu }^{~~\nu } \,A_\nu \big (\Lambda ^{-1}(x-b)\big )\quad , \end{flalign} hence we get from Theorem 2.6 Noether currents and conserved charges that capture the relativistic momentum and angular momentum of the electromagnetic potential. Due to time constraints, we do not study these charges in detail and simply state the main outcome: Since \(A_\mu \) is a multicomponent field transforming in the covector representation of the Lorentz group, one finds that this field carries a nontrivial internal angular momentum, a.k.a. spin, which upon quantization will give rise to spin \(1\) particles, the photons.

The main new feature of electromagnetism, which we didn’t have in any of our previous examples (Klein-Gordon, Dirac, …), is that the action functional (6.11) has a huge (really, really huge!) amount of internal symmetries, given by the transformations

that are labeled by functions \(\alpha (x)\) on the Minkowski spacetime \(\bbR ^d\). Indeed, these transformations leave invariant the field strength tensor

\begin{flalign} F_{\mu \nu } ~\longmapsto ~\partial _\mu (A_{\nu } +\partial _\nu \alpha ) - \partial _\nu (A_{\mu } +\partial _\mu \alpha ) = F_{\mu \nu } + \partial _\mu \partial _\nu \alpha - \partial _\nu \partial _\mu \alpha = F_{\mu \nu } \end{flalign} and also the action

\begin{flalign} S_{\mathrm {MW}}^{}[T_\alpha A] \,=\, S_{\mathrm {MW}}^{}[A] + \int _{\bbR ^d}\partial _\mu \alpha \,j^\mu \,\dd x\, =\, S_{\mathrm {MW}}^{}[A] - \int _{\bbR ^d}\alpha \,\partial _\mu j^\mu \,\dd x\, =\,S_{\mathrm {MW}}^{}[A]\quad , \end{flalign} where in the second step we have integrated by parts and the last step follows from the charge conservation law \(\partial _\mu j^\mu =0\), see (6.2).

Let me emphasize that the main feature here is that \(\alpha \) is not required to be a constant, as it was the case for the internal symmetries of complex Klein-Gordon or Dirac fields, see e.g. Example 2.5. This means that the transformations (6.13) are local symmetries in the sense that they are allowed to act differently at different points \(x\in \bbR ^d\). Such local symmetries are also often called gauge symmetries in the QFT literature and theories that admit gauge symmetries are called gauge theories. Gauge theories, such as electromagnetism and its non-Abelian generalization called Yang-Mills theory, are extremely important in fundamental physics and they are one of the main players in the standard model of particle physics.

As already mentioned above in (6.5), the gauge symmetries (6.13) should not be regarded as some kind of physical symmetries of the system (such as translations or Lorentz transformations), but they are rather a consequence of using a description in terms of variables that has some redundancies. This means that one should consider two potentials that are related by a gauge transformation to be physically equivalent, which is mathematically realized by introducing the equivalence relation

\begin{flalign} A_\mu \,\sim \, (T_{\alpha }A)_\mu = A_\mu + \partial _\mu \alpha \quad . \end{flalign} One way to proceed would thus be to work with the quotient set of all electromagnetic potentials modulo gauge equivalence, but this turns out to be impractical for studying quantization. A more suitable approach is to make use of the gauge symmetries to fix \(A_\mu \) to be of a particularly nice and useful form; this is called gauge fixing. There are different choices of gauge fixings, but the most prominent and (in my personal opinion) most relevant one is given by the Poincaré invariant Lorenz gauge fixing

(Note that here is no typo: Lorenz and Lorentz are two different persons.) It is important to note that every \(A_\mu \) is gauge equivalent to a potential that satisfies the Lorenz gauge fixing condition, hence imposing this gauge fixing does not alter the physical content of the theory. In more detail, given any \(A_\mu \), we define \(\alpha \) by solving the inhomogeneous wave equation \(-\partial ^2 \alpha = \partial _\mu A^\mu \). (It is well-known that this differential equation admits solutions.) Then the gauge transformed potential \((T_\alpha A)_\mu = A_\mu + \partial _\mu \alpha \) satisfies the Lorenz gauge fixing condition \(\partial _\mu (T_\alpha A)^\mu = \partial _\mu A^\mu + \partial ^2 \alpha = 0\). It is important to stress that the Lorenz gauge fixing (6.17) does not fix a unique representative of a gauge equivalence class, hence people call it often a partial gauge fixing. In fact, if \(A_\mu \) satisfies \(\partial _\mu A^\mu = 0\), then so does \((T_\alpha A)_\mu = A_\mu +\partial _\mu \alpha \) for every function \(\alpha \) that satisfies the homogeneous wave equation \(-\partial ^2\alpha =0\). To obtain a full gauge fixing, one can use this residual gauge symmetry in order to fix the value of, say, the \(0\)-component \(A_0\). (For \(j^\mu =0\), one can fix \(A_0=0\), called temporal gauge, which together with (6.17) also implies the Coulomb gauge \(\nabla \cdot \mathbf {A}=0\).) One typically does not make use of such full gauge fixings, because they have the unpleasant feature of being not manifestly covariant under Poincaré transformations, which would considerably complicate QFT computations. However, from the existence of such full gauge fixings, one can draw a useful conclusion: Of the \(d\) components of \(A_\mu \), for \(\mu =0,1,\dots ,d-1\), only \(d-2\) degrees of freedom are nonredundant because \(1\) gets fixed by (6.17) and \(1\) more gets fixed by the additional gauge fixing for \(A_0\). Hence, we see that the electromagnetic potential (modulo gauge symmetry) describes \(d-2\) physical degrees of freedom, which for \(d=4\) are the well-known two independent polarizations of light.

The Lorenz gauge fixing condition (6.17) can be implemented by introducing a Lagrange multiplier \(\xi \) and modifying the action (6.11) according to

\begin{flalign} \label {eqn:MWactionmultiplier} S_{\mathrm {MW}+\mathrm {gf}}^{}[A,\xi ] \,:=\, \int _{\bbR ^{d}} \bigg (-\frac {1}{4}\, F^{\mu \nu }\,F_{\mu \nu } -\frac {\xi }{2}\, (\partial _\mu A^\mu )^2 + A_\mu \, j^\mu \bigg )~\dd x\quad . \end{flalign} The Euler-Lagrange equation for \(A_\mu \) then reads as

\begin{flalign} -\partial ^2 A_\mu + (1-\xi )\,\partial _\mu \big (\partial _\nu A^\nu \big ) \,=\, j_\mu \end{flalign} and the one for \(\xi \) enforces the Lorenz gauge fixing

\begin{flalign} \frac {1}{2} (\partial _\mu A^\mu )^2\,=\,0\quad \Longleftrightarrow \quad \partial _\mu A^\mu \,=\,0\quad . \end{flalign} Fixing \(\xi =1\), one obtains yet another description in terms of the action

\begin{flalign} \nn \widetilde {S}_{\mathrm {MW}}^{}[A] \,&:=\, \int _{\bbR ^{d}} \bigg (-\frac {1}{4}\, F^{\mu \nu }\,F_{\mu \nu } -\frac {1}{2}\, (\partial _\mu A^\mu )^2 + A_\mu \, j^\mu \bigg )~\dd x \\ \nn \,&=\, \int _{\bbR ^{d}} \bigg (-\frac {1}{2} \,\partial ^\mu A^\nu \,\partial _\mu A_\nu +\frac {1}{2}\, \partial ^\mu A^\nu \,\partial _\nu A_\mu -\frac {1}{2}\, (\partial _\mu A^\mu )^2 + A_\mu \, j^\mu \bigg )~\dd x\\ \,&=\,\int _{\bbR ^{d}} \bigg (-\frac {1}{2} \,\partial ^\mu A^\nu \,\partial _\mu A_\nu + A_\mu \, j^\mu \bigg )~\dd x\quad , \end{flalign} where the last step uses twice integration by parts, together with the constraint

\begin{flalign} \partial _\mu A^\mu =0\quad . \end{flalign} Since it is crucial to not forget this constraint when working with this form of the action, let me write it again in a single colored box

The Euler-Lagrange equation corresponding to this action is simply the wave equation (or massless Klein-Gordon equation)

\begin{flalign} \label {eqn:photonwaveequation} -\partial ^2 A_\mu \,=\,0 \end{flalign} for each component \(\mu =0,\dots ,d-1\). This last description is most suitable for discussing the quantization of the electromagnetic potential \(A_\mu \) in the next section. Note that (6.22) is not invariant under all gauge transformations (6.13), but only under the residual gauge transformations preserving the Lorenz gauge fixing, i.e.

\begin{flalign} \label {eqn:Abeliangaugesymmetryresidual} T_\alpha \,:\, A_\mu \,\longmapsto \, (T_\alpha A)_\mu = A_\mu + \partial _\mu \alpha \qquad \text {with}\quad -\partial ^2\alpha =0\quad . \end{flalign}